The efficient numerical solution of the compressible Navier-Stokes equations poses several major computational challenges. In particular, for flow regimes characterized by low Mach number and moderate Reynolds number values, severe time step restrictions may be required by standard explicit time discretization methods.
The use of implicit and semi-implicit methods has a long tradition in low Mach number flows,
see for example the seminal papers [1]}, [2]}, [3]},
as well as many other contributions in the literature on numerical weather prediction,
see e.g. [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]} and the reviews in [13]}, [14]}.
Other contributions have been proposed in the literature on more classical computational fluid dynamics, see e.g. [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}. Many of these contributions focus exclusively on the equations of motion of an ideal gas and their extension to real gases is not necessarily straightforward.
Stability concerns are even more critical in these particular regimes for spatial discretizations based on the Discontinuous Galerkin (DG) method (see e.g. [22]}, [23]} for a general presentation of this method), which is the spatial discretization used in many of the above referenced papers.
